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I'm doing a PhD in Applied Mathematics coming from a Master also in Applied Mathematics. Yet, my undergrad was actually civil engineering, so the first time I actually learned "pure" mathematics was during my Masters, where I took courses such as Real Analysis, Functional Analysis, Measure Theory and Linear Algebra. For this reason, I haven't been exposed to some "basics" that perhaps an undergraduate in math already has. The most blatant example is Abstract Algebra. I'm completely in the dark about the subject, and from time to time it comes up when I'm studying a subject (e.g. Caratheodory's Extension Theorem).

I'm really interested in learning Algebra, but I don't have the time now to take a deep dive. Yet, I feel that knowing the "bare minimum" could help me with other subjects I'm studying. Hence, I was wondering what good references were out there that could serve as a "fast course" on Abstract Algebra. Some of the answers for book recommendations are usually books that are too large. I'd be interested in something that presented the "main" results in a rigorous manner.

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I really like A first course in Abstract Algebra by Fraleigh. He starts by the basics, is very didactic and has nice visualizations. You can get a good grasp on group theory, ring theory and Galois theory. More advanced topics are clearly marked so you can skip them in a first reading.

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I could recommend the book with I was introduced to algebra being almost from scratch: Abstract Algebra by Gregory T. Lee. It is a book with less than 300 pages and in the preface the author talks about what he considers to be the "basic" topics. Gallian's Algebra book is more extensive, but I think you can take it for specific queries and references.

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Alongside with the Fraleigh book you can study the online textbook "Abstract Algebra: Theory and Applications" by Thomas W. Judson. it covers the basics of abstract algebra, including groups, rings, and fields. It is aimed at undergraduate students and includes many examples and exercises.

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